High Grashof number turbulent natural convection on an infinite vertical wall

Abstract

The present study concerns a temporally evolving turbulent natural convection boundary layer (NCBL) adjacent to an isothermally heated vertical wall, with Prandtl number 0.71. Three-dimensional direct numerical simulations (DNS) are carried out to investigate the turbulent flow up to $\textit {Gr}_\delta =1.21\times 10^8$ , where $\textit {Gr}_\delta$ is the Grashof number based on the boundary layer thickness $\delta$ . In the near-wall region, there exists a constant heat flux layer, similar to previous studies for the spatially developing flows (e.g. George & Capp, Intl J. Heat Mass Transfer, vol. 22, 1979, pp. 813–826). Beyond a wall-normal distance $\delta _i$ , the NCBL can be characterised as a plume-like region. We find that this region is well described by a self-similar integral model with profile coefficients (cf. van Reeuwijk & Craske, J. Fluid Mech., vol. 782, 2015, pp. 333–355) which are $\textit {Gr}_\delta$ -independent after $\textit {Gr}_\delta =10^7$ . In this Grashof number range both the outer plume-like region and the near-wall boundary layer are turbulent, indicating the beginning of the so-called ultimate turbulent regime (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56; Grossmann & Lohse, Phys. Fluids, vol. 23, 2011, 045108). Solutions to the self-similar integral model are analytically obtained by solving ordinary differential equations with profile coefficients empirically obtained from the DNS results. In the present study, we have found the wall heat transfer of the NCBL is directly related to the top-hat scales which characterise the plume-like region. The Nusselt number is found to follow $\textit {Nu}_\delta \propto \textit {Gr}_\delta ^{0.381}$ , slightly higher than the empirical $1/3$ -power-law correlation reported for spatially developing NCBLs at lower $\textit {Gr}_\delta$ , but is shown to be consistent with the ultimate heat transfer regime with a logarithmic correction suggested by Grossmann & Lohse (Phys. Fluids, vol. 23, 2011, 045108). By modelling the near-wall buoyancy force, we show that the wall shear stress would scale with the bulk velocity only at asymptotically large Grashof numbers.